The Rouquier Dimension of Quasi-Affine Schemes

We prove that for $X$ a regular quasi-affine scheme of dimension $d$, ${\mathcal{O}}_X$ is a $d$-step generator of $D_{coh}^b(X)$, establishing Orlov’s conjecture in this case. We prove something weaker in the projective case. The main techniques are a spectral sequence argument borrowed from topology and the converse ghost lemma, both suitably adapted to work in this setting. Along the way we prove that on a regular scheme $X$ of dimension $d<\mathrm{\infty }$ any composition of $d+1$ morphisms of $D_{coh}^b(X)$ which are zero on cohomology sheaves is zero.